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Prime Reciprocal Magic Square
A prime reciprocal magic square is a magic square using the decimal digits of the reciprocal of a prime number. Formulation Basics In decimal, unit fractions and have no repeating decimal, while repeats 0.3333\dots indefinitely. The remainder of , on the other hand, repeats over six digits as, 0.\bold42857\bold42857\bold\dots Consequently, multiples of one-seventh exhibit cyclic permutations of these six digits: \begin 1/7 & = 0.1 4 2 8 5 7\dots \\ 2/7 & = 0.2 8 5 7 1 4\dots \\ 3/7 & = 0.4 2 8 5 7 1\dots \\ 4/7 & = 0.5 7 1 4 2 8\dots \\ 5/7 & = 0.7 1 4 2 8 5\dots \\ 6/7 & = 0.8 5 7 1 4 2\dots \end If the digits are laid out as a square, each row and column sums to This yields the smallest base-10 non-normal, prime reciprocal magic square In contrast with its rows and columns, the ''diagonals'' of this square do not sum to ; however, their mean is , as one diagonal adds to while the other adds to . All prime reciprocals in any base with a p - 1 period will generat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Magic Square
In mathematics, especially History of mathematics, historical and recreational mathematics, a square array of numbers, usually positive integers, is called a magic square if the sums of the numbers in each row, each column, and both main diagonals are the same. The "order" of the magic square is the number of integers along one side (''n''), and the constant sum is called the "magic constant". If the array includes just the positive integers 1,2,...,n^2, the magic square is said to be "normal". Some authors take "magic square" to mean "normal magic square". Magic squares that include repeated entries do not fall under this definition and are referred to as "trivial". Some well-known examples, including the #Sagrada Família magic square, Sagrada Família magic square and the #Parker square, Parker square, are trivial in this sense. When all the rows and columns but not both diagonals sum to the magic constant, this gives a semimagic square (sometimes called orthomagic square). ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Princeton University Press
Princeton University Press is an independent publisher with close connections to Princeton University. Its mission is to disseminate scholarship within academia and society at large. The press was founded by Whitney Darrow, with the financial support of Charles Scribner, as a printing press to serve the Princeton community in 1905. Its distinctive building was constructed in 1911 on William Street in Princeton. Its first book was a new 1912 edition of John Witherspoon's ''Lectures on Moral Philosophy.'' History Princeton University Press was founded in 1905 by a recent Princeton graduate, Whitney Darrow, with financial support from another Princetonian, Charles Scribner II. Darrow and Scribner purchased the equipment and assumed the operations of two already existing local publishers, that of the ''Princeton Alumni Weekly'' and the Princeton Press. The new press printed both local newspapers, university documents, '' The Daily Princetonian'', and later added book publishing ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
A096339
A, or a, is the first letter and the first vowel letter of the Latin alphabet, used in the modern English alphabet, and others worldwide. Its name in English is '' a'' (pronounced ), plural ''aes''. It is similar in shape to the Ancient Greek letter alpha, from which it derives. The uppercase version consists of the two slanting sides of a triangle, crossed in the middle by a horizontal bar. The lowercase version is often written in one of two forms: the double-storey and single-storey . The latter is commonly used in handwriting and fonts based on it, especially fonts intended to be read by children, and is also found in italic type. In English, '' a'' is the indefinite article, with the alternative form ''an''. Name In English, the name of the letter is the ''long A'' sound, pronounced . Its name in most other languages matches the letter's pronunciation in open syllables. History The earliest known ancestor of A is ''aleph''—the first letter of the Phoenician ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
223 (number)
223 (two hundred ndtwenty-three) is the natural number following 222 and preceding 224. In mathematics 223 is: *a prime number, *a lucky prime, *a left- truncatable prime, and a left-and-right-truncatable prime. Among the 720 permutations of the numbers from 1 to 6, exactly 223 of them have the property that at least one of the numbers is fixed in place by the permutation and the numbers less than it and greater than it are separately permuted among themselves. In connection with Waring's problem, 223 requires the maximum number of terms (37 terms) when expressed as a sum of positive fifth powers, and is the only number that requires that many terms. See also * The years 223 and 223 BC __NOTOC__ Year 223 BC was a year of the pre-Julian Roman calendar. At the time it was known as the Year of the Consulship of Flaminius and Philus (or, less frequently, year 531 ''Ab urbe condita''). The denomination 223 BC for this year has bee ... References Integers {{Num ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ternary Numeral System
A ternary numeral system (also called base 3 or trinary) has 3 (number), three as its radix, base. Analogous to a bit, a ternary numerical digit, digit is a trit (trinary digit). One trit is equivalent to binary logarithm, log2 3 (about 1.58496) bits of Units of information, information. Although ''ternary'' most often refers to a system in which the three digits are all non–negative numbers; specifically , , and , the adjective also lends its name to the balanced ternary system; comprising the digits −1, 0 and +1, used in comparison logic and ternary computers. Comparison to other bases Representations of integer numbers in ternary do not get uncomfortably lengthy as quickly as in binary numeral system, binary. For example, decimal 365 (number), 365 or senary corresponds to binary (nine bits) and to ternary (six digits). However, they are still far less compact than the corresponding representations in bases such as decimal – see below for a compact way to codi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
59 (number)
59 (fifty-nine) is the natural number following 58 (number), 58 and preceding 60 (number), 60. In mathematics Fifty-nine is the 17th prime number, and 7th super-prime. It is also a good prime, a Higgs prime, an Regular prime#Irregular primes, irregular prime, a Pillai prime, a Ramanujan prime, a Safe and Sophie Germain primes, safe prime, and a Supersingular prime (moonshine theory), supersingular prime, The next prime number is sixty-one, with which it comprises a twin prime. There are 59 stellations of the regular icosahedron. In other fields Fifty-nine is: * The number corresponding to the last minute in a given hour, and the last second in a given minute ** The "59-minute rule" is an informal rule in business, whereby (usually near a holiday) employees may be allowed to leave work early, often to beat heavy holiday traffic (the 59 minutes coming from the rule that leaving one full hour early requires the use of leave, whereas leaving 59 minutes early would not) * The number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary Number
A binary number is a number expressed in the Radix, base-2 numeral system or binary numeral system, a method for representing numbers that uses only two symbols for the natural numbers: typically "0" (zero) and "1" (one). A ''binary number'' may also refer to a rational number that has a finite representation in the binary numeral system, that is, the quotient of an integer by a power of two. The base-2 numeral system is a positional notation with a radix of 2. Each digit is referred to as a bit, or binary digit. Because of its straightforward implementation in digital electronic circuitry using logic gates, the binary system is used by almost all modern computer, computers and computer-based devices, as a preferred system of use, over various other human techniques of communication, because of the simplicity of the language and the noise immunity in physical implementation. History The modern binary number system was studied in Europe in the 16th and 17th centuries by Thoma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Recreational Mathematics
The ''Journal of Recreational Mathematics'' was an American journal dedicated to recreational mathematics, started in 1968. It had generally been published quarterly by the Baywood Publishing Company, until it ceased publication with the last issue (volume 38, number 2) published in 2014. The initial publisher (of volumes 1–5) was Greenwood Periodicals. Harry L. Nelson was primary editor for five years (volumes 9 through 13, excepting volume 13, number 4, when the initial editor returned as lead) and Joseph Madachy, the initial lead editor and editor of a predecessor called ''Recreational Mathematics Magazine'' which ran during the years 1961 to 1964, was the editor for many years. Charles Ashbacher and Colin Singleton took over as editors when Madachy retired (volume 30 number 1). The final editors were Ashbacher and Lamarr Widmer. The journal has from its inception also listed associate editors, one of whom was Leo Moser. The journal contains: # Original articles # Boo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Open Court Publishing Company
The Open Court Publishing Company is a publisher with offices in Chicago and LaSalle, Illinois. It is part of the Carus Publishing Company of Peru, Illinois. History Open Court was founded in 1887 by Edward C. Hegeler of the Matthiessen-Hegeler Zinc Company, at one time the largest producer of zinc in the United States. Hegeler intended for the firm to serve the purpose of discussing religious and psychological problems on the principle that the scientific world-conception should be applied to religion. Its first managing editor was Paul Carus, Hegeler's son-in-law through his marriage to engineer Mary Hegeler Carus.Fields 1992, pg. 138 For the first 80 years of its existence, the company had its offices in the Hegeler Carus Mansion. Open Court specializes in philosophy, science, and religion. It was one of the first academic presses in the country, as well as one of the first publishers of inexpensive editions of the classics. It also published the journals ''Open Court' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Magic Sum
The magic constant or magic sum of a magic square is the sum of numbers in any row, column, or diagonal of the magic square. For example, the magic square shown below has a magic constant of 15. For a normal magic square of order ''n'' – that is, a magic square which contains the numbers 1, 2, ..., ''n''2 – the magic constant is M = n \cdot \frac. For normal magic squares of orders ''n'' = 3, 4, 5, 6, 7, and 8, the magic constants are, respectively: 15, 34, 65, 111, 175, and 260 (sequence A006003 in the OEIS). For example, a normal 8 × 8 square will always equate to 260 for each row, column, or diagonal. The normal magic constant of order ''n'' is . The largest magic constant of normal magic square which is also a: *triangular number is 15 (solve the Diophantine equation where ''y'' is divisible by 4); *square number is 1 (solve the Diophantine equation where ''y'' is even); *generalized pentagonal number is 171535 (solve the Diophantine equation where ''y' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Decimal Place
Significant figures, also referred to as significant digits, are specific digits within a number that is written in positional notation that carry both reliability and necessity in conveying a particular quantity. When presenting the outcome of a measurement (such as length, pressure, volume, or mass), if the number of digits exceeds what the measurement instrument can resolve, only the digits that are determined by the resolution are dependable and therefore considered significant. For instance, if a length measurement yields 114.8 mm, using a ruler with the smallest interval between marks at 1 mm, the first three digits (1, 1, and 4, representing 114 mm) are certain and constitute significant figures. Further, digits that are uncertain yet meaningful are also included in the significant figures. In this example, the last digit (8, contributing 0.8 mm) is likewise considered significant despite its uncertainty. Therefore, this measurement contains f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reciprocals Of Primes
The reciprocals of prime numbers have been of interest to mathematicians for various reasons. They do not have a finite sum, as Leonhard Euler proved in 1737. As rational numbers, the reciprocals of primes have repeating decimal representations. In his later years, George Salmon (1819–1904) concerned himself with the repeating periods of these decimal representations of reciprocals of primes. Contemporaneously, William Shanks (1812–1882) calculated numerous reciprocals of primes and their repeating periods, and published two papers "On Periods in the Reciprocals of Primes" in 1873 and 1874. In 1874 he also published a table of primes, and the periods of their reciprocals, up to 20,000 (with help from and "communicated by the Rev. George Salmon"), and pointed out the errors in previous tables by three other authors. Rules for calculating the periods of repeating decimals from rational fractions were given by James Whitbread Lee Glaisher in 1878. For a prime , the peri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
